Most of the time most employers aren't going to care about "projects" or the like on your resume. They're generally going to be most interested in what you know, know how to do, can well and practically apply, and even better if one's got experience doing so - and better yet if it's practical and relevant experience, and better yet if that's in work/employment contexts.

So, generally mostly appropriately cover the knowledge, skills, etc, and generally giving reasonably accurate information about to what level the knowledge and experience is. And if/when one has various knowledge/skills and the like listed, and they don't see that covered in the employment history, to they extent they're interested or wanting to know where that came/comes from, or what substantiates the claim(s), that's when they'll ask., e.g. "Gee, I see here you have listed that you're quite skilled in foo and have done much with it but I don't see that in your work history, so, how is it you're familiar with foo?" And that's when you've got your opportunity to tell them how you've learned about it, gained experience with it, what you've done with it, etc. - at least as and to the extent they're interested. And also to reasonably cover that, e.g. that was stuff you did on your own because you wanted to learn it and gain more skills and experience with it, or maybe just because it was the most relevant thing to use for doing some cool projects you were highly interested in doing - whatever the background/explanation may be.

suggest some practical projects that would effectively showcase my Linux skills

Gain the skills, list them, but ... projects? Mostly/typically don't list (unless perhaps you otherwise don't have a dang thing you can list in terms of, e.g. work/employment, internships, or post high-school education of any relevance - then maybe list project(s), but otherwise probably not in most cases).

As far as "practical" projects to work on, I'd suggest read through various forums and the like, see the typical questions/issues/problems/challenges/projects sysadmins (of whatever flavor(s) you're intending to target) are doing and working on, troubleshooting and needing help on, etc. Try and well figure out the answers to the issues/challenges/problems they're having. Try to well understand what they're building or have built. As feasible build those things, identify and fix those issues/challenges, etc. That may often be about as close as you'll get to doing the actual work in actual employment environment, and things commonly relevant to those environments.